The spelling changes go farther than just "t" and "tt."
Leonard Clinton Hattaway was my great-grandfather.He had one son and several daughters.After his wife died, he changed the spelling to "Hathaway" but since the son was already married, with a family, the son did not change his name.The son, Carl Clinton Hattaway, was my grandfather.
There are many "Hathaways" in Louisiana now who are descendants from LC and his 2nd wife (Mattie McCormick?) and who are my distant cousins.
As a matter of irony, my father's surname was "TT" and my mother's maiden name was "t."So I am a Hattaway from both sides of my parents.